Uniformly Bounded Orthonormal Bases of Homogeneous Polynomials on the Complex Sphere

Abstract

We resolve the long-standing problem of Rudin concerning uniformly bounded orthonormal bases of homogeneous polynomials on complex unit spheres. Specifically, we show that for every $K\ge 2$ and $N\ge 0$ there exists a basis of degree $N$ homogeneous polynomials in ${\mathbb{C}}^K$ that is orthonormal with respect to the surface measure of the unit sphere of ${\mathbb{C}}^K$ and uniformly bounded by a constant depending only on $K$ on the unit ball of ${\mathbb{C}}^K$. Bourgain resolved the case $K=2$ in 1985 and the case $K=3$ in 2016. No cases $K\ge 4$ were previously known.

38 pages